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CIRCUIT–BASED TRANSIENT MODEL OF GROUNDING ELECTRODE WITH
CONSIDERATION OF SOIL IONIZATION AND CURRENT RATE–OF–RISE
FACTORS
MEHRDAD MOKHTARI
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy (Electrical Engineering)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
APRIL 2016
iii
DEDICATION
To my wife Sepideh, to my daughter Bahar, and to my parents
iv
ACKNOWLEDGEMENTS
First, I would like to express my thanks to my supervisor Professor Dr.
Zulkurnain Abdul–Malek for his comments, guidance, and advices during my
research. I am also grateful to my family for their patience during the entire study.
v
ABSTRACT
The behaviour of a grounding electrode can be predicted by using either the
electrical circuit model or electromagnetic computation. Despite its advantages over
the latter, the grounding circuit model fails to accurately predict the behaviour under
transient conditions due to the absence of two key factors, namely the soil
ionization, and the current rate–of–rise. A new equivalent circuit model of a
grounding electrode with dynamic circuit elements (Rd, Cd, and Ld) was developed to
consider both soil ionization and current rate–of–rise factors. A generalized formula
was derived to calculate the dynamic inductance, Ld, for all standard current wave
shapes such as Conseil International des Grands Réseaux
Électriques (CIGRE), double–exponential, and IEC 62305–1 (International
Electrotechnical Commission). The computed inductance, Ld, dynamically changes
with the change in the lightning current parameters, thus improving its accuracy for
all current rate–of–rise conditions. The consideration for the soil ionization effect
on grounding electrode resistance, Rd, and soil capacitance, Cd, within the equivalent
circuit model was achieved by modelling the soil with a network of two layer
capacitors (TLC) in which soil particles and air voids are the TLC components.
Differential equations were derived to incorporate the soil ionization phenomenon
inside the TLC network. The voltage response of the new equivalent circuit model
and the dynamic circuit elements were determined by using the above–suggested
methods, is more accurate than that of the conventionally determined grounding
circuit models. The overall differences between the equivalent circuit model and
several experiments are 3.3% for the electrode resistance and 2.8% for the electrode
peak voltage. The new equivalent circuit model helps to optimize the overall
grounding electrode design, and to provide a better fast transient protection and
insulation coordination.
vi
ABSTRAK
Tingkah laku elektrod pembumian boleh diramal dengan menggunakan sama
ada model litar elektrik atau menggunakan pengiraan elektromagnetik. Walaupun
mempunyai kelebihan dari yang kedua, model litar pembumian gagal untuk meramal
dengan tepat kelakuan pada keadaan fana disebabkan ketiadaan dua faktor iaitu
pengionan tanah dan juga kadar kenaikan arus. Satu model litar setara yang baru
yang mempunyai unsur litar yang dinamik (Rd, Cd, dan Ld) telah dibangunkan untuk
mengambil kira kedua–dua faktor iaitu pengionan tanah dan juga kadar kenaikan
arus. Satu formula umum telah diterbitkan untuk mengira kearuhan dinamik, Ld,
untuk kesemua bentuk gelombang arus piawai seperti Conseil International des
Grands Réseaux Électriques (CIGRE), eksponen kembar dan IEC 62305–1
(International Electrotechnical Commission). Kearuhan Ld yang dikira secara
dinamiknya berubah dengan perubahan parameter arus kilat, seterusnya ia
meningkatan ketepatan pada kesemua keadaan kadar kenaikan arus. Kesan
pengionan tanah pada perintang elektrod pembumian, Rd, dan kemuatan tanah, Cd, di
dalam model litar setara telah dicapai dengan memodelkan tanah menggunakan
kapasitor dua lapisan (TLC) di mana zarah tanah dan juga lompang udara adalah
komponen TLC. Persamaan pembezaan diterbitkan untuk menggabungkan
fenomena pengionan tanah di dalam rangkaian TLC ini. Sambutan voltan model
litar setara baru di mana elemen litar dinamiknya ditentukan dengan menggunakan
kaedah yang disyorkan di atas, adalah lebih tepat daripada model litar pembumian
konvensional. Perbezaan keseluruhan di antara model litar setara dan beberapa uji
kaji ialah 3.3% untuk rintangan elektrod dan 2.8% untuk voltan puncak elektrod.
Model litar setara baru ini membantu untuk mengoptimumkan reka bentuk elektrod
pembumian secara keseluruhan, dan untuk menghasilkan perlindungan dan
koordinasi penebatan fana pantas yang lebih baik.
vii
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES xii
LIST OF FIGURES xiv
LIST OF SYMBOLS xix
LIST OF ABBREVIATIONS xxi
LIST OF APPENDICES xxii
1 INTRODUCTION 1
1.1 Research Background 1
1.2 Research Problem Statement 5
1.3 Research Objectives 7
1.4 Research Scope 7
1.5 Research Contributions 9
1.6 Thesis Outline 12
2 LITERATURE REVIEW 13
2.1 Introduction 13
2.2 Modeling of Grounding Systems 15
viii
2.2.1 Electromagnetic Model with the Method of Moment 16
2.2.2 Circuit Model 18
2.2.3 Hybrid Model 20
2.2.4 Transmission Line Model 23
2.2.5 Comparison of the Models 26
2.3 Impulse Characteristics of a Grounding Electrode 28
2.4 Impulse Coefficient of a Grounding Electrode 31
2.5 Factors Affecting the Voltage Response of Circuit Model 33
2.5.1 Current Rate–of–Rise Factor 34
2.5.1.1 Current Waveform Characteristics 34
2.5.1.2 Voltage Response Analysis of the Lumped
Circuit Model 40
2.5.1.3 Voltage Response Analysis of the
Transmission Line Model 44
2.5.2 Soil Ionization Factor 47
2.5.2.1 Transient Soil Characteristics 48
2.5.2.2 Soil Conduction Mechanism 48
2.5.2.3 Thermal Process 49
2.5.2.4 Ionization Process 52
2.5.2.5 Soil Critical Electric Field Value, Ec 54
2.5.2.6 Influence of Water Content and
Temperature on Ec 55
2.5.2.7 Influence of Grain Size on Ec 57
2.5.2.8 Influence of Soil Compaction on Ec 58
2.5.2.9 Soil Ionization Effect on Voltage Response
of the Grounding Electrode 59
2.6 A Critical Review on Soil Ionization Modeling 61
2.6.1 Characteristics of Grounding Electrode Resistance 61
2.6.2 Soil Ionization Models 63
2.6.2.1 CIGRE Model 63
2.6.2.2 Bellaschi Model 64
2.6.2.3 Mohamad Nor Model 66
2.6.2.4 Liew–Darveniza Model 68
ix
2.6.2.5 Accuracy Analysis of Soil Ionization
Models 72
2.7 Summary 77
3 METHODOLOGY 78
3.1 Introduction 78
3.2 Equivalent Circuit Model of Grounding Electrode 80
3.3 Determination of Dynamic Circuit Element Values 81
3.3.1 Dynamic Resistance, Rd, and Capacitance, Cd 81
3.3.1.1 Soil Ionization Model 81
3.3.1.2 Two–layer Capacitor (TLC) Network 83
3.3.1.3 Ionization in TLC Network 90
3.3.1.4 Determination of Required Parameter
Values for Soil Ionization Modeling 93
3.3.1.5 Equivalent Circuit Model of Soil Ionization 96
3.3.1.6 Energy Balance Concept 98
3.3.1.7 Arc Resistance Computation 99
3.3.1.8 Equations to Determine the Dynamic
Grounding Electrode Resistance and Soil
Capacitance 105
3.3.2 Dynamic Inductance, Ld 107
3.3.2.1 Circuit Response of the Double–
Exponential Current Waveform 111
3.3.2.2 Circuit Response of the CIGRE Current
Waveform 112
3.3.2.3 Circuit Response of the IEC Current
Waveform 112
3.4 Validation of the Equivalent Circuit Model 113
3.5 Performance Evaluation of the Equivalent Circuit Model 115
3.5.1 Methodology of the Hybrid Model 117
3.5.1.1. Electric Field Computations 118
3.5.1.2. Equivalent Radius and Grounding
Electrode Resistance 119
x
3.5.2 Methodology for Incorporation of Soil Ionization in
CDEGS Software 120
3.5.2.1 Step 1: Specifying the System
Configuration 121
3.5.2.2 Step 2: Computing the Current in an
Electrode 122
3.5.2.3 Step 3: Computing the Electric Field 122
3.5.2.4 Step 4: Computing the Equivalent Radius 122
3.5.2.5 Step 5: Recomputing the Electric Field 123
3.5.2.6 Computation of the Equivalent Radius 123
3.6 Summary 126
4 RESULTS AND DISCUSSIONS 127
4.1 Introduction 127
4.2 Development of the Equivalent Circuit Model 128
4.3 Validation of the Equivalent Circuit Model 130
4.3.1 Validation with Bellaschi’s Experimental Work and
Liew–Darveniza’s Theoretical Model 131
4.3.2 Discussion on the Results of Validation with
Bellaschi’s Experimental Work and Liew–
Darveniza’s Theoretical Model 137
4.3.3 Validation with Geri’s Experimental Work and
Theoretical Model 138
4.3.4 Discussion on the Results of Validation with Geri’s
Experimental Work and Theoretical Model 140
4.3.5 Validation with Grcev’s Electromagnetic
Computational Model 140
4.3.6 Discussion on the Results of Validation with Geri’s
Experimental Work and Theoretical Model 145
4.4 Performance Evaluation of the Equivalent Circuit Model 146
4.4.1 Equivalent Circuit Model Setup 146
4.4.1.1 Lightning Current Specifications 147
4.4.1.2 Grounding Electrode Specifications 148
xi
4.4.1.3 Required Parameters for Soil Ionization
Modeling 149
4.4.2 Simulation Results 151
4.4.2.1 A Sample Electrode Configuration: 10–m–
long Electrode Buried in 200–Ω.m Soil 152
4.4.2.2 Eight Other Electrode Configurations 157
4.4.2.3 Summary of Key Parameters 157
4.4.3 Analysis on the Simulation Results 159
4.4.3.1 Influence of Electrode Length and Soil
Resistivity on Grounding Electrode
Resistance 159
4.4.3.2 Influence of Current Parameters on
Grounding Electrode Impedance 162
4.4.3.3 Influence of Soil Resistivity on Grounding
Electrode Voltage Response 165
4.4.3.4 Influence of Energy Balance on Grounding
Electrode Resistance 166
4.4.3.5 Influence of Energy Balance on Grounding
Electrode Voltage Response 167
4.4.4 Comparison of the Results with other Work 169
4.5 Summary 171
5 CONCLUSIONS AND FURTHER WORK 172
5.1 Conclusions 172
5.2 Recommendations for Future Work 174
REFERENCES 176
Appendices A–C 187–207
xii
LIST OF TABLES
TABLE NO. TITLE PAGE
2.1 Typical rock–soil permittivity values 20
2.2 Summary of the grounding system transient modeling 28
2.3 Critical breakdown strength for soils with different
grain sizes [102] 58
2.4 Critical breakdown strength of soils with different
densities [74] 59
2.5 The specifications of the grounding electrode from the
experiment [10] 72
2.6 The resistances computed by the models and their
errors compared to the experimental value 74
2.7 The peak voltage values computed by the models and
their errors compared to the experimental value 76
3.1 Type of soil based on soil resistivity and resistance of a single
rod [107] 95
3.2 Size range of soil particles based on British standard
BS 5930 [108] 95
4.1 Grounding electrode specifications for ground types F, M, and
K [10] 132
4.2 Comparison between the resistance and peak voltage
values obtained from the equivalent circuit model (EC)
and experimental work performed by Bellaschi (B) 132
4.3 Comparison of the results obtained from the equivalent
circuit model (EC) and Liew–Darveniza’s model (LD)
xiii
with experimental work performed by Bellaschi for
ground types F, M, and K 135
4.4 The specifications and current parameters of the slow–
fronted current 147
4.5 The specifications and current parameters of the fast–
fronted current 147
4.6 The specifications of the grounding electrode 149
4.7 Required parameters for soil ionization modeling 150
4.8 Condition of grounding electrode and main parameter
values obtained under slow–fronted current 150
4.9 Condition of grounding electrode and main parameter
values obtained under fast–fronted current 151
4.10 Identification of the eight electrodes presented in
Appendix B 157
4.11 Transient values of grounding electrodes under slow–
fronted current 158
4.12 Transient values of grounding electrodes under fast–
fronted current 159
4.13 Comparison between resistance, impedance, and voltage
obtained under slow–fronted current by the equivalent
circuit model (EC), CIGRE model, hybrid model, and
CDEGS software with soil ionization consideration 170
4.14 Comparison between resistance, impedance, and voltage
obtained under fast–fronted current by the equivalent
circuit model (EC), CIGRE model, hybrid model, and
CDEGS software with soil ionization consideration 171
xiv
LIST OF FIGURES
FIGURE NO. TITLE PAGE
2.1 A representation of critical review flow on transient
grounding system 15
2.2 Representation of a typical grounding electrode under
lightning current by an equivalent lumped circuit model 19
2.3 Distributed model of a grounding electrode system. Vi
and ii correspond to the voltage and current at the
respective section. VLi is the inductive voltage at the
respective section [7] 25
2.4 Ratio of harmonic impedance to low frequency
resistance for the horizontal electrodes with (a) 1–m
length, (b) 10–m length, and (c) 30–m length [9] 30
2.5 CIGRE concave current waveform [17] 37
2.6 IEC 62305–1 current waveform [17] 39
2.7 Current steepness versus time [17] 40
2.8 Illustration of the grounding electrode under impulse
current 41
2.9 Current waveforms with 10–kA amplitude and 1–µs
front time 42
2.10 CIGRE and double–exponential current derivatives 42
2.11 The voltage response of the lumped circuit model of a
horizontal 15–m–long grounding electrode at the point
of applied current [17] 43
xv
2.12 Impulse current distribution (iLi) illustrated for the first
three sections of the 15–m–long grounding electrode and
relevant leakage currents (iR–Ci) 45
2.13 Voltage distribution illustrated for the first three sections
of the 15–m–long grounding electrode. VR’–C’i and VLi
are the resistive–capacitive and inductive voltages,
respectively 45
2.14 Voltage response of the distributed circuit with 15, 30,
45, and 60 sections 47
2.15 Breakdown initiation time of soil for different
conductivities (the continuous lines are the best–fit
curves) [91] 49
2.16 Current responses to an applied 30 kV, 2.5 µs voltage of
(a) large grounding grid system of the utility substation,
and (b) 10 m by 10 m grounding grid system [93] 51
2.17 Soil ionization propagation model [11] 52
2.18 Electrical discharge in a void inside a dialectic material
and its equivalent circuit [95] 52
2.19 Critical electric field values obtained based on Oettle
and Manna for moderate soil moisture (εg=10) 55
2.20 Influence of water content on the critical impulse
breakdown field strength of soil at different temperature
[101] 56
2.21 Influence of temperature on the critical impulse
breakdown field strength of soil at different water
content [101] 57
2.22 Computed components of the potential at the injection
point of current of a 10–m–long electrode in earth with
ρ=100 Ω.m [9] 60
2.23 Transient potential rise at the current injected point of
the grounding grid system [58] 60
2.24 Variation of grounding electrode resistance under
impulse current discharge [10] 62
xvi
2.25 Hysteresis characteristics of grounding electrode
resistance [10] 62
2.26 Equivalent circuit model of soil ionization proposed by
Mohamad Nor et al. [11] 67
2.27 Soil characteristics under impulse current in a
hemispherical model for a direct sparking connection
[14] 69
2.28 The resistivity profile of the soil ionization zone in the
Liew– Darveniza’s model [14] 69
2.29 Resistance variation of the grounding electrode 73
2.30 Hysteresis characteristics of the grounding electrode
resistance 74
2.31 Voltage response of the grounding electrode 75
3.1 Research methodology flow 79
3.2 An equivalent circuit model of a grounding electrode
with dynamic elements 80
3.3 Microstructure photo of soil [74] 82
3.4 A network of soil particles and air voids to represent a
portion of soil medium. The grey and white cells
represent the soil particle and the air void, respectively 82
3.5 A portion of soil modeled with a network of soil
particles and air voids under impulse current (a) without
ionization and (b) with ionization occurrence in air voids 83
3.6 Physical dimensions and electrical characteristics of a
typical TLC 84
3.7 A representation of a portion of soil with a network of
two–layer capacitors (TLCs) 85
3.8 Waveforms of key voltages in a typical TLC 91
3.9 Variations of the electric field in soil particle and air
inside a typical TLC 92
3.10 Soil ionization circuital model under impulse condition 97
3.11 V–I characteristics of the arc for different current rate–
of–rise values. The current rate–of–rise increases in
curves I to III 102
xvii
3.12 A grounding electrode with a length l and radius a
buried in a uniform soil with an injected impulse current 118
3.13 Construction of a mesh with the size of m×n to
determine the nodes 118
3.14 Summary of the developed five–step method 121
3.15 Representations of the (a) constructed mesh in the lower
region of the horizontal electrode to determine the
electric field, E, in the defined nodes nmn and (b) typical
electric field variation along a typical profile, Pc 124
3.16 Variation of ionization distance, Pc(t)×DBP, and
equivalent radius, aeq(t), for electrode with time 125
4.1 Developed circuit model of grounding electrode with the
consideration of soil ionization and current rate–of–rise
factors 128
4.2 The flow of the equivalent circuit model 129
4.3 Resistance variation and voltage response of the
grounding electrodes obtained by using the equivalent
circuit model, Bellaschi’s experiment, and Liew and
Darveniza’s computational model in (a) Ground F, (b)
Ground M, and (c) Ground K 133
4.4 Hysteresis characteristic of the grounding electrodes
obtained by using the equivalent circuit model,
Bellaschi’s experiment, and Liew and Darveniza’s
computational model in (a) Ground F, (b) Ground M,
and (c) Ground K 136
4.5 Comparison of voltage responses of a horizontal
electrode obtained by the equivalent circuit model,
Geri’s experiment, Geri’s model, and Grcev’s model 139
4.6 Comparison of voltage responses of a vertical electrode
obtained by the equivalent circuit model, Geri’s
experiment, Geri’s model, and Grcev’s model 139
4.7 Typical slow– and fast–fronted current waveforms 142
xviii
4.8 Voltage responses of the 10–m–long electrode in soil
with resistivity of 100 Ω.m under (a) slow– and (b) fast–
fronted currents obtained by the equivalent circuit model
and by the Grcev’s electromagnetic model 143
4.9 Voltage responses of the 30–m–long electrode in soil
with resistivity of 100 Ω.m under (a) slow– and (b) fast–
fronted currents obtained by the equivalent circuit model
and by the Grcev’s electromagnetic model 144
4.10 Representation of (a) current waveforms and (b) current
derivatives 148
4.11 Transient characteristics of grounding electrode with l =
10 m, ρ = 200 Ω.m under slow–fronted current 154
4.12 Transient characteristics of grounding electrode with l =
10 m, ρ = 200 Ω.m under fast–fronted current 156
4.13 Variation of the grounding electrode resistance versus
soil resistivity obtained at current maximum 160
4.14 Variation of the grounding electrode resistance versus
electrode length obtained at current maximum 161
4.15 Grounding electrode impedance under slow–fronted
current 163
4.16 Grounding electrode impedance under fast–fronted
current 164
4.17 Grounding electrode impedance versus soil resistivity 165
4.18 Grounding electrode resistance at t = Th 167
4.19 Grounding electrode voltage at t = Th 168
xix
LIST OF SYMBOLS
a – Physical electrode radius
aeq – Equivalent electrode radius
L – Electrode inductance
n – Maximum number of consecutive TLCs along
which soil ionization occurs
q – Electric charge
Q – Total electric charge
R – Low current and low frequency grounding electrode
resistance
x – Soil ionization length
aeq – Equivalent radius
Aarc – Area of the arc channel
Cd – Dynamic capacitance
Cg – Soil capacitance of non–ionized region
C1i – Soil capacitance of ionized region
da – Length of air void
ds – Length of soil particle
dTLC – Length of TLC
Ea – Electric field in air void
Earc – Electric field intensity in arc channel
Es – Electric field in soil particle
Ja – Current density in air void
Js – Current density in soil particle
kµ – Electron mobility
Ld – Dynamic inductance
xx
Pa(t) – Instantaneous heat power absorbed by the soil
Pb(t) – Instantaneous heat power balance in the arc channel
Pp(t) – Instantaneous heat power produced in the arc channel
Qb – Heat energy balance
Rd – Dynamic resistance
Rg – Soil resistance of non–ionized region
RG(t) – Dynamic grounding electrode resistance
R1i–arc – Arc resistance
R1i–soil – Soil resistance of ionized region
Tbr – Time to breakdown
Tf – Current front time
Th – Current half value
ti – Required time for air breakdown
tu – Sufficient voltage breakdown time
Ua – Voltage across the air void
Us – Voltage across the soil particle
Usa – Voltage across a TLC
v(t) – Grounding electrode voltage at current injection point
xmax – Maximum soil ionization length
Z1i(t) – Equivalent impedance of the parallel elements
R1i–soil(t), R1i–arc(t), and C1i(t)
Zg – Equivalent impedance of the parallel elements
Rg and Cg
ε – Permittivity
σ – Conductivity
ρ – Soil resistivity
τ – Time constant
εs – Soil permittivity
εa – Air permittivity
σs – Soil particle conductivity
σa – Air conductivity
σarc – Arc conductivity
σarc(t) – Time dependent arc conductance
νd – Electron drift velocity
xxi
LIST OF ABBREVIATIONS
CDEGS – Current Distribution, Electromagnetic Fields,
Grounding, and Soil Structure Analysis
CIGRE – The International Council on Large Electric Systems
(in French: Conseil International des Grands Réseaux
Électriques, abbreviated CIGRÉ)
DBN – Distance Between Nodes
DBP – Distance Between Points
DFS – Distance From Surface
EM – Electromagnetic Approach
EM–MoM – Electromagnetic Approach with Method of Moment
EMTP – Electromagnetic Transient Program
FFT – Fast Fourier Transform
IEC – The International Electrotechnical Commission
IEEE – The Institute of Electrical and Electronics Engineers
IFFT – Inverse Fast Fourier Transform
MATLAB – Matrix Laboratory
MoM – Method of Moment
PSCAD – Power System Computer Aided Design
TLC – Two–Layer Capacitor
xxii
LIST OF APPENDICES
APPENDIX TITLE PAGE
A Numerical Computations of a 10–m–long Grounding
Electrode 187
B Transient Characteristics of the Grounding Electrodes 189
C List of Publications 205
CHAPTER 1
INTRODUCTION
1.1 Research Background
Grounding electrodes are used to efficiently disperse the high amplitude
currents in the event of power system faults or lightning strikes. In addition, they
are also important to ensure low and safe ground voltage levels are maintained. A
typical configuration of a single grounding electrode is a buried horizontal electrode
(counterpoises) or a driven vertical electrode (rod). The current that flows through
these electrodes, especially due to lightning strikes, has a fast rate–of–rise and
together with the effect of soil resistivity, it dictates the characteristic of the
grounding electrode as either inductive or capacitive. This characterization is a
significant factor in determining the overall grounding electrode impedance. The
grounding electrode impedance can be modeled under transient conditions by using
circuit models [1–5]. In the circuit models, the grounding electrode is represented
either as a distributed or as a lumped circuit [6] with R, C, and L elements. In the
lumped circuit model, the circuit elements are combined together into one section to
give only a single resistance, inductance, and capacitance to represent the whole
electrode impedance. On the other hand, in the distributed model, the elements are
uniformly (as per–unit–length parameters) or non–uniformly distributed along the
electrode. The elements of the circuits are computed by using relevant formulae
2
proposed by Sunde [7] and Dwight [8]. According to Sunde formula, the resistance
and capacitance of the grounding electrode are related. In other words, R need to be
first obtained before the value of C is determined. However, inductance, L, is
determined independently. These circuit elements (R, C, and L) have static rather
than dynamic values, that is they do not change during the impulse current duration.
Hence, the effect of current amplitude and current rate–of–rise are not taken into
account when computing the voltage response. Soil ionization is a phenomenon
which occurs when impulse current is dispersed in the soil. This is especially true
when the amplitude of the discharge current is high. The phenomenon substantially
affects the values of R and C. In addition, the impulse current rate–of–rise also has a
significant effect on the electrode inductance. Consequently, because of the soil
ionization and current rate–of–rise are not taken into account [9] when computing R,
C, and L, their static values are therefore overestimated. Hence, the computed
voltage response of the grounding electrode using those static values is erroneous
and sometimes this error can be significantly large and require further attention.
Various soil ionization models were proposed to improve the accuracy of
computed R and C. These include work by Bellaschi et al. [10], Mohamad Nor, et
al. [11], CIGRE [12], Geri [13], and Liew and Darveniza [14]. An attempt to include
the soil ionization mechanism is described by Geri in [13]. In addition, the author
also describes electric field enhancement in soil when subjected to high impulse
current discharge, which in turn causes the breakdown of air voids that do exist
among the mostly solid soil particles. The air breakdown is usually described in the
form of arcs and arc growth. The arc growth is also usually further described as
either in continuous form or discrete (or stepped) form. A discrete type of arc or
ionization growth occurs because of the fact arcs cease to grow when the electric
field intensity becomes less than the critical value, Ec. It is to be noted that previous
soil ionization models assume a continuous or diffused ionization growth.
Furthermore, the models incorporate the soil ionization effect by modifying either
the soil resistivity or the grounding electrode radius. However, this modification is
not a valid technique when considering the above–mentioned soil ionization
mechanism.
3
It is also to be noted that, the majority of the soil ionization models proposed
in [10–13] assume that the grounding electrode resistance is only a function of
discharge current. Therefore, these models fail to determine the transient grounding
electrode resistance during the current decay (tail) period. This is because in current
decay period, the value of the grounding electrode resistance is mainly dependent on
the energy balance between the produced heat energy in the arc within the voids and
the absorbed heat energy by the soil particles. The so–called energy balance concept
is defined as a concept where the computation of the air void arc resistance (and
hence the grounding electrode resistance) is obtained by computing the actual
balanced heat energy transferred between the air voids and the soil solid particles. A
detailed explanation of this concept is given in the methodology section of this
thesis.
In the grounding models developed by Bellaschi et al. [10], Mohamad Nor,
et al. [11], CIGRE [12], and Geri [13], the arc resistance is assumed to be equal to
zero. Therefore, previous soil ionization models fail to characterize the relationship
between grounding electrode resistance and impulse current, in particular the
hysteresis characteristic. It is known that neglecting the hysteresis characteristic
causes a large error in the computation of the grounding electrode resistance
especially when the impulse current reduces during the decaying period. According
to [103], compared to the experimental value, the grounding electrode resistances
obtained by models proposed by CIGRE [12], Bellaschi et al. [10], and Mohamad
Nor et al. [11] at the current half time (Th) give errors of 20%, 17%, and 25%,
respectively.
It is also known that the computation of resistance, R, and capacitance, C, are
related. Hence, the computation of C becomes erroneous when R is not accurate.
Among the previous soil ionization models, only the soil ionization model proposed
by Liew and Darveniza [14] gives an adequate accuracy when computing the
transient grounding resistance. This is because the energy balance between the arc
and the bulk of soil is considered. Nevertheless, the Liew–Darveniza’s model still
has several shortcomings. For example, the model assumes a diffused rather than a
4
discrete ionization growth. Hence, the discrete–breakdown path, which exists
because of an air breakdown in voids enclosed among the soil particles, cannot be
modeled [14, 15]. Furthermore, the solution for the general expressions of the soil
resistivity (with soil ionization effect) often results in numerical divergence [16].
Another deficiency of the Liew–Darveniza’s model is that the effect of soil
capacitance is not considered. It is known that neglecting the soil capacitance leads
to inaccurate grounding electrode voltage [9, 17].
In addition, to overcome the deficiency caused by inductance value, L, on
electrode voltage, two methods were previously proposed, namely, the constant and
the length–dependent distribution of parameters along the electrode. According to
these methods the simultaneous effect of inductance value and current rate–of–rise
factor on electrode voltage (v = L di(t) / dt) is reduced by distributing the inductance
along the electrode. However, the results obtained from the above–mentioned
methods are only valid under slow–fronted current waves (that is, the front time, Tf
> 1µs). Incorrect voltage responses are still obtained when the circuits are under
fast–fronted current waves (Tf < 1 µs).
Overall, the accuracy of the previously proposed models is dependent on
several key parameters of the grounding electrode and the impulse current. The key
parameters are defined as the electrode length, the current amplitude, the current
front time, and the soil resistivity.
It is concluded that the soil ionization and current rate–of–rise factors have
significant effects on both the circuit element values and the voltage response of the
grounding electrode. However, these two factors are not properly considered in
many grounding circuit models.
5
1.2 Research Problem Statement
A major drawback of both lumped and distributed grounding circuit models
is that they fail to produce the correct transient voltage at the injection point of the
lightning current. The root–cause of this error is due to the static nature and
inaccurate estimation of R, C, and L, which are computed without considering two
important influencing factors, namely, the soil ionization and current rate–of–rise.
Although several soil ionization models were previously proposed to
enhance the value of R, but they still have several shortcomings. Firstly, in the
previous models, the effect of soil ionization is only indirectly considered on the soil
resistivity and on the electrode radius rather than the preferred direct effect on the
grounding electrode resistance itself. Secondly, the previous soil ionization models
can only be used for continuous type of ionization growth rather than the preferred
discrete type of ionization growth, which frequently occurs when the grounding
electrode is subjected to high amplitude impulse currents. Thirdly, apart from the
soil ionization model proposed by Liew and Darveniza, all previous soil ionization
models are inaccurate in determining the grounding electrode resistance because
they neglect the effects of two important aspects, namely, the arc resistance and the
so–called energy balance concept. Even though the soil ionization model proposed
by Liew and Darveniza can be considered as accurate, the proposed model is
complicated and the general expressions given to compute the variation of the soil
resistivity often result in numerical divergence.
The effect of current rate–of–rise plays a direct role in determining the
inductance, L. Two previous methods, namely, the constant, and the length–
dependent distribution of parameters along the electrode, have been proposed to
compute L. Both methods do not provide a correct electrode voltage response under
fast front current waves (that is, the front time, Tf < 1 µs). Under such current
waveforms, correct electrode voltage response can only be obtained using a
dynamically variable inductance depending on the front time of the waveform. It is
6
obvious that the effect of current rate–of–rise must be directly considered when
computing the inductance L.
It is important to consider the soil ionization and current rate–of–rise factors
in designing and implementing the power system protection and safety. This is
because these factors directly affect the resultant grounding electrode voltage when
discharging high current impulses. For example, the electrode peak voltage
significantly reduces when the soil ionization is considered. Consequently, a direct
improvement in the grounding electrode performance is achieved [9]. According to
[13], for a typical horizontal grounding electrode, a 66.5% reduction in electrode
voltage is observed when the soil ionization is considered in the computation of the
grounding electrode resistance and voltage. Therefore, by considering the soil
ionization effect on the behavior of the grounding electrode when discharging high
current impulses, the margin of the protection level in power system can be
increased. On the contrary, the high current rate–of–rise factor together with the
electrode inductance cause the peak voltage of the grounding electrode to increase
considerably. The peak voltage computed by the previously proposed grounding
circuit model is considerably higher because of neglecting the current rate–of–rise
factor. According to [9] a difference of 26% in the peak voltage of a typical
grounding electrode was observed when computed using the electromagnetic
computational model and the circuit model. Therefore, an overestimation may exist
when designing the power system insulation coordination including the ratings of
the protective devices. In short, by considering both the soil ionization and the
current rate–of–rise factors, the cost of power system insulation coordination
implementation can be reduced and hence the economic benefit of such
considerations.
In this thesis, several new methods of computation are developed and
proposed to enhance the accuracy of the above–mentioned static R, C, and L circuit
elements. The key contribution to the success of the developed methods is the
incorporation of two additional factors namely, 1) the effect of soil ionization (for
improved R and C), and 2) the consideration of current rate–of–rise (for improved
7
L), in a new equivalent circuit model of a grounding electrode with dynamic
elements for transient conditions.
1.3 Research Objectives
The objectives of the study are as follows:
(i) To develop a new equivalent circuit model of a grounding electrode with
dynamic resistance, capacitance, and inductance elements, which are
achieved by considering the soil ionization and current rate–of–rise
factors.
(ii) To validate the accuracy of the equivalent circuit model of a grounding
electrode by comparing the results obtained from the model with those
obtained from other well–known experimental work and theoretical
models.
(iii) To evaluate the performance of the equivalent circuit model of a
grounding electrode by changing the key parameters of the grounding
electrode and impulse current.
1.4 Research Scope
In this research, an equivalent circuit model of a grounding electrode is
developed to model the grounding electrode with the consideration of soil ionization
and current rate–of–rise factors. The effect of soil ionization is directly considered
8
in the form of a dynamic electrode resistance. Similarly, a dynamic soil capacitance
is also proposed using Sunde equation (RC = ρε). As for the dynamic inductance,
Ld, a new generalized formula is also derived.
MATLAB codes and CDEGS software are used to compute the dynamic
resistance, capacitance, and inductance of the equivalent circuit model. When
CDEGS software is used as a part in the computational step, all assumptions made
in the corresponding electromagnetic approach with the method of moments are
accepted as correct. For example, the electrode is assumed as a thin wire to presume
a zero current at the open ends of an electrode. In addition, the grounding electrode
is assumed to be made of cylindrical metallic conductor at which the ratio of the
length of the conductor segment to its radius is larger than one.
In the performance analysis of the model, the soil critical electric field value,
Ec, is considered as 300 kV/m, as suggested by IEEE standard. In the validation
process, the results obtained from the model are compared to those obtained from
the widely known theoretical models and experimental works. In particular, the
following ranges of parameters are used: 40 Ω.m ≤ ρ ≤ 5000 Ω.m, 3 m ≤ l ≤ 30 m, Im
≤ 30 kA, Tf > 0.15 µs, d ≥ 0.5 m, and Ec ≥ 70 kV/m, where ρ is the soil resistivity,
Im is the current amplitude, Tf is the current front time, l is the electrode length, d is
the burial depth, and Ec is the soil critical electric field.
The application of the new equivalent circuit model is limited to single
horizontal electrode (counterpoise) or single vertically driven rod. Furthermore, the
new equivalent circuit model is only valid for homogeneous and uniformed soil.
Finally, the voltage response of the grounding electrode model is computed at the
current injection point, which is usually at one end of the said electrode.
9
1.5 Research Contributions
The significant contributions of the study are as follows.
i) Critical Review on Previous Models
Several lumped and distributed grounding circuit models were previously
proposed to characterize the grounding electrode impedance behavior under
transient conditions. However, the previous models fail to accurately
determine the electrode voltage. In this study, previous models were
critically reviewed to determine the root–causes of error in determining the
electrode voltage response. A critical and comprehensive review is
presented in Chapter 2. A review on the previously proposed circuit models
of grounding electrodes revealed that neglecting two factors, namely, the soil
ionization and current rate–of–rise factors, substantially affect the accuracy
of the circuital models to determine the electrode voltage under slow– and
fast–fronted currents. The review had enabled the development of a new and
more accurate equivalent circuit model for a grounding electrode.
ii) A New Equivalent Circuit Model for Grounding Electrode with
Improved Accuracy
Previous transient models for a grounding electrode are inaccurate and
require further improvements. An innovative and accurate equivalent circuit
model of a grounding electrode with consideration of the key factors of soil
ionization and current rate–of–rise was developed. In the equivalent circuit
model of the grounding electrode, dynamic resistance, Rd, capacitance, Cd,
and inductance, Ld, were used to characterize the grounding electrode
resistance, soil capacitance, and electrode inductance, respectively. New
models and methods were developed to determine the above–mentioned
dynamic circuit elements. A new soil ionization model was developed to
determine the grounding electrode resistance and soil capacitance. The
10
principle of two–layer capacitor (TLC) was taken into account to model the
soil particle and air void. Differential equations were derived to incorporate
the soil ionization phenomenon inside the TLC network. A new method
based on dynamic and static characteristics of arc and so–called energy
balance concept was developed to compute the arc resistance in soil. Finally,
a set of formulae were derived to compute the dynamic grounding electrode
resistance and dynamic soil capacitance with soil ionization effect.
The significance of the developed soil ionization model are: 1) the dynamic
grounding electrode resistance and dynamic soil capacitance values were
obtained by considering the soil ionization effect, and 2) the hysteresis
characteristic of the grounding electrode resistance was achieved. A new
generalized formula was derived to calculate the dynamic inductance value
of the grounding electrode with the consideration of the current rate–of–rise
factor. The significance of the derived formula is that the dynamic
inductance is accurately determined for all standard current wave shapes,
such as CIGRE, double–exponential, and IEC 62305–1.
iii) Validation and Comparative Data for the Model
Several well–known experimental works and theoretical models were used to
validate the accuracy of the new equivalent circuit model for a grounding
electrode. The results from the experimental work are more accurate. The
specifications and characteristics of the grounding electrode and impulse
current defined in the several experimental works and theoretical models
were used to set up the equivalent circuit model of the grounding electrode.
The voltage responses and grounding electrode resistance values obtained
from the equivalent circuit model were compared to those obtained by the
experimental works and theoretical models. The comparison of the results
with other theoretical models shows that the equivalent circuit model gives a
better performance and accuracy in terms of voltage waveform, peak voltage,
and grounding electrode resistance value. In addition, the grounding
11
electrode resistance shows a comparable hysteresis characteristic compared
to experimental ones.
iv) Evaluation and Performance Data of the Model
The performance of the equivalent circuit model was evaluated by changing
key parameters values of the grounding electrode and impulse current
(electrode length, current amplitude, current front time, and soil resistivity).
It is noted that the range of values are stated in the scope of this thesis. The
simultaneous effect of soil ionization and current rate–of–rise were taken into
account. The time variation of the grounding electrode resistance, arc
resistance, hysteresis characteristic, grounding electrode impedance as well
as the voltage of the electrodes were obtained. The small differences
between the values obtained from the equivalent circuit model and those
from theoretical models show the excellent performance of the equivalent
circuit model in the above–mentioned challenging conditions. The proposed
equivalent circuit model can be used as to provide reliable and more accurate
results when computing grounding electrode response to various injected
currents.
v) Optimized Grounding Electrode Design
The main significance of the study is the improved accuracy of the new
equivalent circuit model of a grounding electrode with dynamic elements of
Rd, Cd, and Ld. The elements can be obtained by simultaneous consideration
of soil ionization and current rate–of–rise factors. This research shows that
the voltage response of the improved model is very accurate and comparable
to the other theoretical and experimental results. The new equivalent circuit
model can be used to obtain the voltage response of a grounding electrode in
typical installations and hence helps to optimize the overall grounding
electrode design due to the improved accuracy. This new model also
indirectly addresses any safety concern arising from such grounding
electrode design especially when subjected to fast transients.
12
vi) Improved Insulation Coordination
The obtained results can be used as a reliable source for validation of any
grounding electrode model. Another important significance of the model is
that the equivalent circuit model can be directly applied or connected to
power system equipment in standard simulation platforms. In this way, an
accurate grounding electrode effect on the transient performance of key
power equipment such as surge arresters can be obtained. Using this
integrated approach, a better protection and insulation coordination
characteristics can be designed.
1.6 Thesis Outline
Chapter 1 gives mainly emphasis to the objectives of the study and the
methodology used to solve the stated problems. Chapter 2 presents a critical review
on related works conducted to model the grounding electrodes and highlights the
existing problems of the models. Chapter 3 presents a methodology used to develop
a new grounding electrode model. Chapter 4 is assigned to validate and evaluate the
accuracy and the performance of the equivalent circuit model by comparing the
results obtained from the equivalent circuit model with those obtained from the
well–known experimental works and theoretical models. Finally, Chapter 5 presents
the conclusions and future recommendations.
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